We define the function
\(\begin{split}i\colon [0,\infty ) \times S_pN \rightarrow \mathbf {N}, \quad i(t,v) \text{ is the index of the index form of $\gamma _{p,v}$ on the interval $[0,t]$.}\end{split}\) 
By this we mean that \(i(v,t)\in \mathbf {N}\)  is the dimension of the vector space \(V(t,v)\)  of piecevise smooth vector fields \(X,Y\)  along the geodesic \(\gamma _{p,v}\)  that vanishes at 0, \(t\)  and for which the bi-linear symmetric form
\(I_{t,v}(X,Y):=\int _{0}^{t}\langle \mathrm {D}_s X, \mathrm {D}_s Y\rangle _g+\langle R(\dot{\gamma }_{p,v},X)\dot{\gamma }_{p,v}, Y \rangle _g \; \mathrm {d}s,\) 
is negative semi-definite. By the proof of the Morse index theorem (see [1]}) it holds that
\(i(t,v)=\left\lbrace \begin{array}{ll}0, & t<{\tau _{\text{con}}}(p,v)\\\delta (v), & t\in [{\tau _{\text{con}}}(p,v), \varepsilon (v)),\end{array}\right.\) 
where \(\varepsilon (v)>0\)  depends on \(v \in S_pN\) . Moreover, no \(\gamma _{p,v}(t)\) , for \(t \in ({\tau _{\text{con}}}(p,v), \varepsilon (v))\)  is conjugate to \(p\)  along \(\gamma _{p,v}\) .
We choose \(t \in ({\tau _{\text{con}}}(p,v_0), \varepsilon (v_0))\) . By the proof of the Morse index theorem, the index form \(I_{t,v_0}\)  can only be negative semi-definite on a finite dimensional vector space \(\mathcal {V}_-(0,t)\) , which is independent to the geodesics that are close to \(\gamma _{p,v_{0}}\) . Since \(\gamma _{p,v_0}(t)\)  is not conjugate to \(p\)  along \(\gamma _{p,v_0}\)  it follows from [1]} that \(I_{t,v_0}(Y,Y)>0\)  for vector fields \(Y\)  in a sub-space \(W_{t,v_0}\subset \mathcal {V}_-(0,t)\) , which by the Finite-dimensional spectral theorem satisfies
\(\mathcal {V}_-(0,t)=W_{t,v_0}\oplus V_{t,v_0}.\) 
Moreover, the linear operator \(L_{t,v_0}\colon \mathcal {V}_-(0,t) \rightarrow \mathcal {V}_-(0,t) \) , corresponding to the bi-linear form \(I_{t,v_0}\) , has \(i(t,v_0)\)  negative and \(\dim \mathcal {V}_-(0,t)-i(t,v_0)\)  positive eigenvalues. Since these eigenvalues depend continuously of the initial direction \(v \in S_pN\) , that are near \(v_0\) , we can find a neighborhood \(U\subset S_pN\)  of \(v_0\)  such that the linear operator \(L_{t,v}\) , related to the bi-linear form \(I_{t,v}\) , has \(i(t,v_0)\)  negative and \(\dim \mathcal {V}_-(0,t)-i(t,v_0)\)  positive eigenvalues for every \(v \in U\) . Furthermore, the continuity of the first conjugate time function, that was established in Lemma REF , gives \({\tau _{\text{con}}}(p,v)<t\)  for \(v \in U\) .
By the proof of the Morse index theorem the function \(i(t,v)\)  is increasing in \(t\) . We have in particular that
\(1\le \delta (v)=i({\tau _{\text{con}}}(p,v),v)\le i(t,v)= i(t,v_0)= \delta (v_0)=1, \quad \text{for every $v \in U$}.\) 
This ends the proof.
